Half-Life and Decay Constant

Half-life and decay constant are the same fact in two dialects. The decay constant λ is the fundamental one: it is the probability per unit time that any individual nucleus, chosen at random and regardless of its history, will decay. That single assumption — a constant per-nucleus probability — produces the exponential law $N = N_0 e^{-\lambda t}$ and nothing else. The half-life is what you get by asking that law when $N/N_0 = \tfrac{1}{2}$: take logarithms and $t_{1/2} = \ln 2/\lambda$, with ln 2 = 0.6931471806. Large λ, short half-life. The two numbers carry identical information, and which one a source quotes is a matter of discipline rather than physics.

Cobalt-60 has a half-life of 5.27 years, so $\lambda = 0.6931/5.27 = 0.1315$ per year, or $4.17 \times 10^{-9}$ per second once the years are converted. That per-second figure is the useful one, because activity is $A = \lambda N$. One gram of Co-60 holds $6.022\times10^{23}/60 = 1.00\times10^{22}$ nuclei, so its activity is $4.17\times10^{-9} \times 1.00\times10^{22} \approx 4.2\times10^{13}$ Bq — about 42 TBq, or 1100 curies, from a single gram. λ is what converts a count of atoms into a radiation hazard.

A third quantity lives in the same family and is the one most often mixed up with the others. The mean lifetime τ — the average time an individual nucleus survives before decaying — is simply $1/\lambda$, which makes it $t_{1/2}/\ln 2 = 1.443\,t_{1/2}$. The mean life is 44% longer than the half-life, and they are not interchangeable. Particle physics tables usually quote τ, nuclear medicine and health physics usually quote $t_{1/2}$, and a value copied across that boundary without conversion is wrong by a factor you will not notice from the magnitude alone. The same structure appears wherever a fixed fractional loss per unit time does — drug clearance, capacitor discharge through a resistor, the attenuation of light through an absorbing medium.

The unit trap is mechanical and unforgiving: λ and $t_{1/2}$ are reciprocals, so their units must be reciprocals too. A λ quoted per year set against a time measured in seconds is off by $3.16\times10^7$. Decide on one time unit at the start and hold it through the whole problem.

One property of λ is worth stating because it is genuinely unusual. It does not depend on temperature, on pressure, on the chemical compound the atom sits in, or on how long the nucleus has already existed. Essentially every chemical rate constant doubles or better for a 10 K rise; λ does not move. Heating a radioactive sample, dissolving it, or bonding it into a molecule changes nothing measurable, because the decay is a nuclear event and chemistry happens far outside the nucleus. In living systems, though, a second clock runs alongside: a radiopharmaceutical is also excreted, and the two combine as $1/T_{eff} = 1/T_{phys} + 1/T_{bio}$, so the effective half-life in a patient is always shorter than the physical one.